Magnetic Field Extrapolations And Current Sheets

1
Magnetic Field Extrapolations And Current Sheets

• B. T. Welsch,1 I. De Moortel,2 and J. M.
  McTiernan1
• 1Space Sciences Lab, UC Berkeley
• 2School of Mathematics Statistics,
• University of St. Andrews, Scotland

2
Abstract

• Solar flares and coronal mass ejections (CMEs)
  --- phenomena which impact our society, but are
  scientifically interesting in themselves --- are
  driven by free magnetic energy in the coronal
  magnetic field.
• Since the coronal magnetic field cannot be
  directly measured, modelers often extrapolate the
  coronal field from the photospheric magnetograms
  --- the only field measurements routinely
  available.
• The best extrapolation techniques assume that
  the field is force free (coronal currents
  parallel the magnetic field), but that currents
  are not simply a linear function of the magnetic
  field.
• Recent tests (Metcalf et al. 2007), however,
  suggest that such non-linear force-free field
  (NLFFF) extrapolation techniques underestimate
  free magnetic energy.
• We hypothesize that, since relaxation-based
  NLFFF techniques tend to smooth field
  discontinuities, such approaches will fail when
  current sheets are present.
• Here, we test this hypothesis by applying the
  Optimization NLFFF method (Wheatland et al. 2000)
  to two configurations from an MHD simulation ---
  one with strong current concentrations, and one
  with weak concentrations.
• This work is supported by a NASA Sun-Earth
  Connections Theory grant to SSL/UCB.

3
Free energy is the energy difference between the
actual and potential field energies.

• For a given field B, the magnetic energy is
• U ? ? dV (B B)/8?.
• The lowest energy the field could have would
  match the same boundary condition Bn, but would
  be current-free (curl-free), or potential
• B(P) - ??, with ?2? 0, and U(P) ? ? dV B(P)
  2/8?
• The difference U(F) U U (P) is the energy
  available to power flares and CMEs!

4
Non-linear force-free field (NLFFF)
extrap-olations give B, allowing integration of
B2/8?.

• Strictly, NLFFF extrapolation should not be
  applied to non-force-free photospheric
  magnetograms.
• Wheatland et al. (2000) described the
  Optimization Method to determine a NLFFF,
  B(x,y,z), that matches a given magnetic boundary
  condition.
• McTiernan has implemented this extrapolation
  procedure in IDL, and distributed it via SSW, in
  the NLFFF package.

5
The Optimization Method minimizes an objective
function, L
Specification of B is required on all surfaces
for solar applications, magnetograms give B(x,y)
at z0, and B(P) is used on other boundaries.
6
L can be expressed as a functional of ?t B,
providing a way to update B to minimize L.
and
7
The Optimization Method for NLFFF extrapolation
has been tested in several ways.

• Wheatland et al. (2000) and Schrijver et al.
  (2006) used the analytic solution of Low Lou
  (1990).
• Abbett et al. (2004) used MHD simulations from
  Magara et al. (2001).
• Metcalf et al. (2007) used a hybrid potential /
  non-potential reference model.

8
The Optimization method performed well with
Magaras (2004) MHD simulations in tests by
Abbett et al. (2004).
Potential
Actual
NLFF
Figure 1.
Chromosphere
Photosphere
9
Tests by Metcalf et al. (2007), however, showed
free energy is underestimated from photospheric
magnetograms.
Model E/Epot
Optimization Methods
From Metcalf et al. 2007
10
Why does the Optimization method fail at
estimating free energy?

• The non-force-free character of the boundary
  clearly plays a role. (Chromospheric
  extrapolations do better.)
• But also the Optimization method is a relaxation
  method, and might relax away current
  concentrations, as seen by Antiochos et al. 1999.
  
• (These currents store free energy.)

11
Antiochos et al. (1999) found that NLFF
extrapolations missed current sheets in their
breakout simulations, and therefore
underestimated free energy.
12
We decided to test the Optimization method
against MHD configurations with current sheets.
Figure 2.
INITIAL STATE
FINAL STATE
From De Moortel Galsgaard (2006)
13
We extrapolated two MHD states, Figure 2s final
state.
Figure 3.
Current density in pink.
14
and an intermediate state.
Figure 4.
Current density in pink density is lower than
final state shown in Figure 3.
15
The NLFF extrapolations were initiated from
potential fields.

• Figure 5. Initial conditions for attempted
  extrapolation of field in Fig. 3 (left) and Fig.
  4 (right).

16
The NLFF extrapolations did not accurately
reproduce the topology of the MHD configurations.

• Figure 6. NLFF extrapolations for configurations
  in Figure 3. (left) and 4 (right).

17
Conclusions

• Its a good idea to start the research described
  in ones SPD Abstract sooner than the week before
  the meeting.
• (CONCLUSIONS ARE PRELIMINARY!)
• 2. Considering 1., operator error (by Welsch!) is
  not unlikely.
• 3. When the initial topology is far from the
  actual topology, Optimization Method
  extrapolations fail.

18
Future Work

• We plan to include trial smoothing penalty
  terms in L, to prevent smoothing away current
  concentrations.
• We plan to try a weighting function, w(x,y,z)
  (Wheatland et al. 2000, Wiegelmann et al. 2004),
  to limit the effect of forces on boundaries,
  e.g.,

19
References

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